L9n9

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L9n8.gif

L9n8

L9n10.gif

L9n10

Contents

L9n9.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n9 at Knotilus!

L9n9 is 9^2_{60} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n9's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X7,14,8,15 X15,18,16,5 X11,16,12,17 X17,12,18,13 X13,8,14,9 X2536 X4,9,1,10
Gauss code {1, -8, 2, -9}, {8, -1, -3, 7, 9, -2, -5, 6, -7, 3, -4, 5, -6, 4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9n9 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^3+t(2)^3-t(2)^2-t(1) t(2)+t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}-\frac{1}{q^{11/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +2 z a^9+2 a^9 z^{-1} +z a^7-z^5 a^5-5 z^3 a^5-5 z a^5-a^5 z^{-1} (db)
Kauffman polynomial -z^6 a^{12}+5 z^4 a^{12}-6 z^2 a^{12}+2 a^{12}-z^7 a^{11}+5 z^5 a^{11}-6 z^3 a^{11}+3 z a^{11}-a^{11} z^{-1} -2 z^6 a^{10}+11 z^4 a^{10}-15 z^2 a^{10}+5 a^{10}-z^7 a^9+6 z^5 a^9-10 z^3 a^9+7 z a^9-2 a^9 z^{-1} -z^6 a^8+6 z^4 a^8-8 z^2 a^8+3 a^8+z^3 a^7-z a^7+z^2 a^6-a^6-z^5 a^5+5 z^3 a^5-5 z a^5+a^5 z^{-1} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-10χ
-4         11
-6         11
-8      11  0
-10     1    1
-12    131   1
-14   1      1
-16   11     0
-18 11       0
-20          0
-221         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4 i=-2
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}_2 {\mathbb Z}^{3} {\mathbb Z}
r=-3 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}_2 {\mathbb Z}
r=-1
r=0 {\mathbb Z} {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L9n8

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L9n10